Before the late Thursday afternoon huddle in the Longworth Building, Carter told reporters that the group was on track to release a bill by the first week of June. The group had been working to reach agreement on how the legislation would handle health care for immigrants on the pathway to citizenship. If members don’t agree on that element, they would separate it out of the comprehensive bill and vote on it separately.

Carter had said the group has agreed to present separate proposals — Democratic and Republican — on a guest-worker program for low-skilled workers. A source said after the meeting that the two sides were still preparing to release their own plans on guest workers.

The health care component was the main topic at a private meeting of the negotiators Thursday evening.

Still, Carter insisted that the legislation would be “95 percent” bipartisan.

“I’m encouraged that we may have a meeting of the minds tonight,” Carter said before the meeting. “But one way or the other, I’m not going to be sitting in a room, going through the motions and not solving any problems anymore.”

Johnson was unable to attend the meeting as he was undergoing a “common surgery” to treat a hernia, a spokeswoman said. Labrador, who was also not able to attend, joined the meeting by phone.

Thursday’s developments were a hopeful sign for the group, one day after Republicans were threatening to halt negotiations and only hours after House Speaker John Boehner expressed concern the group is imploding.

Before the meeting, Gutierrez had declined to go into specifics of the negotiations, but signaled optimism for an agreement.

“Nothing has shaken that confidence in the last 24 hours,” Gutierrez told reporters. “If anything, I am strengthened in my conviction that we can do this.”

The breakthrough in the House came as the Senate Judiciary Committee continued to revise and debate the immigration legislation agreement brokered by a bipartisan group of negotiators in that chamber. But the bill — even if it ultimately passes the Senate with wide margins — is likely to generate significant conservative opposition in the GOP-led House.

Carter said he vehemently opposes the Senate bill, and declared it dead on arrival in the House.